The liquid and vapor phases in particle models with Kac potentials

We consider here the model introduced in \cite{LMP} by Lebowitz, Mazel and Presutti to study phase transitions for point particles in the continuum. The mean field phase diagram of the model in the $(\beta,\la)$-plane, $\la$ the chemical potential, $\beta$ the inverse temperature, consists of a smooth curve $\la=\la(\beta)$, $\beta>\beta_c>0$, where two phases (liquid and vapor) coexist, elsewhere the phase is unique. In \cite{LMP} it has been proved that phase transitions (in the sense of non uniqueness of DLR measures) persist when the mean field interaction is replaced by a Kac potential with small but fixed scaling parameter $\ga$ (mean field being derived in the limit $\ga\to 0$). In particular it is shown that for any $\beta>\beta_c$, there is a phase transition at $\la=\la(\beta,\ga)$ for any $\ga$ small enough with $\la(\beta,\ga)\to\la(\beta)$ as $\ga\to 0$. Here we complete the analysis of the phase diagram by showing that if $\la\ne \la(\beta)$, then, for all $\ga$ small enough, there is a unique DLR measure.